EXPONENT INEQUALITIES FOR THE BULK CONDUCTIVITY OF A HIERARCHICAL MODEL

The bulk conductivity sigma*(p) of the bond lattice in Z(d) is considered, where the bonds have conductivity 1 with probability p or epsilon greater-than-or-equal-to 0 with probability 1 - p. Various representations of the derivatives of sigma*(p) are developed. These representations are used to analyze the behavior of sigma*(p) for epsilon = 0 near the percolation threshold p(c), when the conducting backbone is assumed to have a hierarchical node-link-blob (NLB) structure. This model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds. Exact asymptotics of d(2)sigma*/dp2 for the NLB model are proven under some technical assumptions. The proof employs a novel technique whereby d(2)sigma*/dp2 for the NLB model with epsilon = 0 and p near p(c) is computed using perturbation theory for sigma*(p) (for two- and three-component resistor lattices) around p = 1 with a sequence of epsilon's converging to 1 as one goes deeper in the hierarchy. These asymptotics establish convexity of sigma*(p) (for the NLB model) near p(c), and that its critical exponent t obeys the inequalities 1 less-than-or-equal-to t less-than-or-equal-to 2 for d = 2, 3, while 2 less-than-or-equal-to t less-than-or-equal-to 3 for d greater-than-or-equal-to 4. The upper bound t = 2 in d = 3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model.